Efficient Branch and Bound Algorithm for The Dynamic Layout Problem A thesis presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirement for the Degree Master of Science Anish Jariwala November, 1995 ACKNOWLEDGMENTS I am indebted to my thesis advisor, Dr. Thomas A. Lacksonen for providing constant support, advice and encouragement throughout this work. I have really enjoyed the pleasant association that I have had with him over the last couple of years. I would also like to thank my thesis committee members Dr. Robert Williams and Dr. Brian Manhire for their critical review of my work. My parents and sister have my greatest love and admiration and I deeply appreciate the sacrifices they have made on my behalf. Simply saying thanks is not enough. Finally, I would like to thank my friends who have made studying at Ohio university an enjoyable experience. Table of Contents Acknowledgments Table of Contents List of Tables List of Figures Chapter 1. Introductions 2. Literature Review 2.1 Purpose of Literature Review 2.2 Classification of Facility Layout Problems 2.3 Discrete Representation of Area 2.4 Continuous Representation of Area 2.5 Overview and Comparisons 3. Problem Statement 3.1 Mixed Integer Linear Program Model 3.2 Non-overlapping Constraints 3.3 Research Goals 4. Approach and Preliminary Testing 4.1 Introductions 4.2 Branch and Bound Algorithm 4.2.1 Stopping Criteria for Optimization 4.2.2 Using a Threshold 4.3 Preliminary test Problems 4.4 Built-in Cplex Preprocessing Rules 4.5 Node Selection Strategy 4.6 Initial Solution 4.6.1 Setting Branching Direction 4.6.2 Comparison of Different Initial Solution Algorithms 4.7 Variable Selection Strategy 4.8 Setting Priority Orders 4.9 Branching Direction Strategy 4.10 Conclusion 5. Results 5.1 Experimental Design 5.2 Small Problems 5.3 Large Problems 5.4 Selection of Best Approach 5.5 Comparison 5.6 Summary 6. Conclusions and Recommendations 6.1 Conclusions 6.2 Future Recomrnendati.ons References Appendices Abstract List of Tables Tables 4.1 Preliminary Test Problems 4.2 Initial Solution Test Problems 4.3 Comparison between two starting solutions 5.1 Small problems 5.2 Large problems 5.3 ANOVA for small problems 5.4 ANOVA for large problems 5.5 Comparison of number of nodes for small problems 5.6 Comparison of cost for large problems 5.7 Comparison of nodes for large problems List of Figures Figure Page 3.1 Coordinates of department 3.2 Separation in the x-direction 3.3 Separation in the y-direction 5.1 Design of Experiment Matrix: 5.2 Effect of threshold setting on nodes for small problems 5.3 Effect of threshold setting on cost for small problems 5.4 Effect of priority setting on nodes for small problems at 0.00 threshold 5.5 Effect of threshold setting on. cost for large problems 5.6 Effect of branching direction on cost at 0.80 threshold for large problems 5.7 New MIP-Mixed Integer Programming Algorithm for DLP 5.8a Block diagram for datal23a-time period 1 5.8b Block diagram for datal23a-time period 2 5.8~Block diagram for datal23a-time period 3 Chapter 1 Introduction A company should strive to reduce indirect costs. Material handling constitutes a major part of indirect costs in a facility. ranging from 10% to 80% of the indirect cost of facility (Tompkins and White, 1984). Therefore even small improvements in material handling costs makes a large reduction in total indirect costs. The cost of material flow is a function of the distance the material is moved between divisions called departments in a manufacturing facility. To reduce material handling costs, it is essential to have an optimal arrangement of departments to minimize the total distance traveled. According to a survey done by 1rJicole & Hollier (1983), the average life of any layout is 2.6 years. Hence, it is necessary to consider the dynamic nature of facility layout. Other factors such as new technology, new product lines and a market-based dynamic environment also force one to think about dynamic facility layout. Material handling costs are no longer constant over the planning horizon, creating the need for radical layout changes. The Dynamic lalyout problem (DLP) finds not only the present layout but a series of layouts for the future by taking into account total flow costs and rearrangement costs. 1.1 Facility Layout The problem of facility layout is to decide the proper positions of a collection of 2 departments to minimize total flow cost. It is a specific application of the assignment problem. Facility layouts also include architectural space planning, manufacturing cell layout and VLSI design, etc. Tlie type of facilities varies in each case but the problem formulation remains the same. Hillier and Lieberman (1990) refer to the assignment problem as resources being allocated to activities on a one-to-one basis. Only one department is allocated to one and only one location and vice-versa. The main objective of the facility layout problem is to minimize overall cost, which is directly related to material flow between departments. Generally material flow is represented by the product of the amount of material and the distance the material is moved. The distance traveled is estimated using rectilinear distance between centroids of the departments since material moves through a facility by way of aisles. Constraints ensure departments do not overlap and departments are of proper size. 1.2 Dynamic Facility Layout (DLP) A large number of optimal and heuristic algorithms have been designed and published to solve the facility layout problems. They are mostly static in nature. A DLP generates a series of static layouts each representing the facility layout at a unique point of time. The DLP minimizes the cost of changing from one layout to another, called the relayout cost, plus the flow cost of each layout. 1.3 Problem Statement An existing program can solve static as well as dynamic layout problems. The current program takes a considerable amount of time and memory to obtain optimal solutions. The greater the number of departments, the higher will be the solution time and memory required. This thesis develops a more efficient program which reduces solution time, reduces memory required, and improves total cost where optimal solution is not reached. Chapter 2 Literature Review 2.1 Purpose of Literature Review The purpose of the literature review is to explore current research in different facility layout problems, including various models to represent the facility layout problem and algorithms that solve these models. 2.2 Classification of Facility Layout Problems There are mainly two ways of representing areas of departments in the facility layout problem: discrete and continuous. In the discrete representation of area, the total area required by all the departments is divided into blocks of equal size. The algorithms which use this representation either prespecify the dimensions of the departments or represent each department as a set of grid blocks. This often leads to irregularly shaped departments and inaccurate representation of areas. Continuous representation of areas overcome the drawbacks of discrete representation. It allows unequal department areas to be represented accurately by the coordinates of their comers- upper right and lower left. Constraints are used to limit their length-to-width (aspect) ratio. 2.3 Discrete Representation of Area This section discusses models and algorithms that allow a discrete representation of unequal areas. In this case there are two different types of algorithms-heuristic and optimal. 2.3.1 Heuristic Algorithm A heuristic algorithm works towards an optimal solution but ends its search when it finds a 'good enough' solution. As computation increases, these algorithms will approach the optimal solution. The purpose of the heuristic algorithm is not to find the best or optimal solution but to find an acceptable solution in an acceptable amount of time using an acceptable amount of computer memory. 2.3.la Construction Algorithms ALDEP (Seehof and Evans, 1967) randomly selects a department and assigns it to the upper left comer of the layout. The next department selected for the assignment is one which has highest closeness rating in the relationship (REL) chart to the initial department and places it in the layout. The closeness ratings are qualitative ratings signifying the desirability of placing two departments adjacent to each other. The above process is repeated until all the departments are placed in the layout. Several layouts can 6 be generated because the initial department is chosen randomly. ALDEP has the ability to handle up to 63 departments and up to three floors. Some of the disadvantages of this algorithm are that the total area requirement for the floor must be known, the departments areas are restricted to integer multiple of blocks and layouts generated may contain unacceptably irregular shaped departments. CORELAP (Lee and Moore, 1967) also uses a total closeness rating of each department to determine a layout. Unlike A.LDEP which randomly selects the first department to be assigned, CORELAP selects the .first department depending upon its total closeness rating value. The selected first department is placed in the center of the layout and the rest of the departments are added subsequently depending on their relationships to the departments already assigned. This algorithm also suffers the same disadvantages as ALDEP, such as inaccurate representation of departmental areas and irregularly shaped departments which requires manual adjustment to enhance the practical feasibility of layout. 2.3.lb Improvement Algorithms The most popular improvement algorithm is CRAFT, which was developed by Armour and Buffa (1963).